Explicit 3-colorings for exponential graphs Adrien Argento and - PowerPoint PPT Presentation

Explicit 3-colorings for exponential graphs Adrien Argento and Alantha Newman Universit´ e Grenoble-Alpes November 14, 2018 1

The Hypercube 11010 11110 10110 Vertices: All binary strings of length n . Edges: Edge between two vertices (strings) if they differ in exactly one po- sition. 2

The Hypercube is Bipartite How can we find a bipartition? Use Breadth-First-Search. This takes O (2 n ), but is polynomial in size of hypercube. Now, suppose we are given the vertices one-by-one by an adversary. Then can we assign each vertex to a side of the bipartition in time poly( n )? 3

Bipartition of the Hypercube How can we find a bipartition? Use Breadth-First-Search. This takes O (2 n ), but is polynomial in size of Hypercube. Now, suppose we are given the vertices one-by-one by an adversary. Then can we assign each vertex to a side of the bipartition in time poly( n )? All vertices with even number of 1’s in EVEN, and All vertices with odd number of 1’s in ODD. 4

“Explicit” Bipartition of the Hypercube 11010 11110 10110 We can determine to which side a vertex belongs in time O ( n ). Explicit bipartition: reason why a vertex belongs to one side or the other. Now we describe another bipartite graph ... First, a definition. 5

The 3-Coloring Exponential Graph K • Let C n denote odd cycle on n nodes (i.e., n is odd). are all 3-colorings of C n (i.e., 3 n vertices). • Vertices of K = K C n 3 • Note that 3-colorings can be non-proper. 6

The 3-Coloring Exponential Graph K • Let C n denote odd cycle on n nodes . • Vertices of K = K C n are all 3-colorings of C n . 3 • Add an edge between two 3-colorings if the bipartite graph between them is properly colored. 7

∀ ij ∈ E ( C n ), check if i ′ j ′′ and i ′′ j ′ are properly colored. The categorical graph product: C n × K 2 . Edge. 8

∀ ij ∈ E ( C n ), check if i ′ j ′′ and i ′′ j ′ are properly colored. The categorical graph product: C n × K 2 . No Edge. 9

The 3-Coloring Exponential Graph K • Let C n denote odd cycle on n nodes . • Vertices of K = K C n are all 3-colorings of C n . 3 • Add an edge between two 3-colorings if the bipartite graph between them is properly colored. • K is not bipartite, but we now describe an induced subgraph that is. 10

Definition of Fixed Points A “fixed point” means color of that node in a neighboring vertex is “fixed”. 11

Induced Subgraph that is Bipartite (First Rule) • Vertices of K = K C n are all 3-colorings of C n . 3 (1) Keep vertices with an even number of fixed points. (Call this the even component of K .) 12

Induced Subgraph that is Bipartite (Second Rule) • Vertices of K = K C n are all (i.e. 3 n ) 3-colorings of C n . 3 (1) Keep vertices with an even number of fixed points. (2) Fix any edge e in C n and remove all vertices of K corresponding to colored copies of C n in which edge e is monochromatic. • Call this graph K e . • Theorem: K e is bipartite [El-Zahar and Sauer 1985]. 13

Application of K e Being a Bipartite Graph We can 3-color the even component of K C n 3 . Why do we want to do this ? When χ ( H ) > k , we can k -color K H iff Hedetniemi’s Conjecture is k true [El-Zahar and Sauer 1985]. They also showed: The problem of 3-coloring K H 3 when χ ( H ) > 3 can be reduced to: The problem of 3-coloring the even component of K C n 3 . 14

Application of K e Being a Bipartite Graph Goal: 3-color the even component of K C n 3 . Algorithm: [Tardif 2006] Let e = ab . Find bipartition of K e . If vertex in K e belongs to LEFT side, color is color ( a ). If vertex in K e belongs to RIGHT side, color is color ( b ). If vertex has c = color ( a ) = color ( b ), then color is c . 15

3-Coloring the Even Component of K C n 3 . For any edge e , copies of C n in which e is monochromatic form hitting set for the odd cycles in the even component of K C n 3 . 16

Tardif’s Question on Explicit Colorings Is there is an “explicit bipartition” for K e ? [Tardif 2006] Problem: Given a 3-colored copy of C n such that: 1. there is an even number of fixed points, and 2. e is not monochromatic, Assign this copy to one of the sides of the bipartition in time poly( n ). 17

An Explicit Bipartition for K e Compute the label ℓ of a vertex: { 01 , 12 , 20 } = +1 , { 10 , 21 , 02 } = − 1. Suppose label is sum of all arcs in “chord cycle”. Here, ℓ = 0. 2 2 2 2 0 0 2 2 1 1 0 2 0 2 2 2 1 1 a a b b All vertices in the same connected component have same label. 18

An Explicit Bipartition for K e Compute the label ℓ of a vertex: { 01 , 12 , 20 } = +1 , { 10 , 21 , 02 } = − 1. Suppose label ℓ = 0. Recall e = ab . Then “petit chemin” from a to b is either majority Red or majority Blue . 2 2 0 1 2 0 2 2 1 a b a b All vertices in the same component have same label. 19

An Explicit Bipartition for K e Compute the label ℓ of a vertex: { 01 , 12 , 20 } = +1 , { 10 , 21 , 02 } = − 1. Suppose label ℓ > 0 (e.g., ℓ = 6). Then “petit chemin” from a to b is either > ℓ 2 or < ℓ 2 . 2 0 0 0 1 2 0 1 0 a b a b All vertices in the same component have same label. 20

An Explicit Bipartition for K e Rule: 1. If p ( a, b ) > ℓ/ 2, LARGE side. 2. If p ( a, b ) < ℓ/ 2, SMALL side. Main challenge: show that two neighbors f and g have p f ( a, b ) and p g ( a, b ) values that are anti-correlated: ℓ − 1 ≤ p f ( a, b ) + p g ( a, b ) ≤ ℓ + 1 . Proof idea: Prove by induction that p f ( a, b ) and P g ( b, a ) are correlated. 21

Conclusions + Open Question Assume χ ( H ) ≥ 4 and let f ( H ) denote a 3-colored copy of H . If f ( H ) is a vertex of a 3-chromatic component of K H 3 , we can assign a color to this vertex in time O ( | H | ). If f ( H ) is an isolated vertex of K H 3 , we can determine this in time O ( | H | ). If f ( H ) is a vertex of a bipartite component of K H 3 , then we can assign a color in time O ( | H | ) · W , where W is the number of vertices in bipartite components that we have seen so far (i.e., time is “input sensitive”). Question: Given 3-colored copy of H , determine if this copy contains an odd cycle with an even number of fixed points. 22